Mechanical simulator for downhole pumping systems



Dec. 28, 1954 F. w. BUBB 2,698,133

MECHANICAL SIMULATOR FOR DOWNHOLE PUMPING SYSTEMS Filed Jan. 8, 1951 2 Sheets-Sheet l F/G. 2. BY Y Hwf AT RNE Dec. 28, 1954 F. w. BUBB 2,698,133

MECHANICAL SIMULATOR FOR DOWNHOLE PUMPING VSYSTEMS Filed Jan. 8, 1951 2 Sheets-Sheet 2 INVENTOR.

F.W. BUBB 7 A A7' ORNEVS United States Patent C) MECHANICAL SIlVIULATOR FOR DOWNHOLE f PUMPING SYSTEMS Frank W. Bubb, Webster Groves, Mo., assignor to Phillips Petroleum Company, a corporation of Delaware Application January 8, 1951, Serial No. 204,927

13 Claims. (Cl. 23S-61) This invention relates to a simulator which is a mechanical analogue of a pumping system. In one specific aspect, 1t relates to a simulatorwhich is a mechanical analogue of the tubing, sucker rod, oil column, and pump in a deep well pumping unit.

A typical deep well system includes a series or string of metal rods, referred to in the art as sucker rods, these rods extending through tubing positioned in the well, the lower end of the tubing carrying a pumping unit. The tubmg, in turn, is suspended within a well casing and, under some conditions, the lower portion of the interspace between the tubing and casing may contain a column of oil. At the top of the well, the sucker rod string extends through a stuing box and is driven by a prime mover, such as an electric motor or an internal combustion engine, through a flywheel or bull wheel driven by the prime mover. This flywheel, in turn, drives a crank or counterbalance which is coupled through a walking beam to the top end of the sucker rod string.

As the pumping operation is carried out, a column of oil rises in the region between the tubing and the sucker rod string, thus producing a viscous drag both upon the tubing and sucker rod string. The weight and elasticity of the sucker rod string, as well as the tubing and oil column, produce elastic strains in the pumping unit so that the sucker rod string and tubing behave in a manner somewhat analogous to elongated springs.

This elastic movement of the sucker rod string and tubing oftentimes results in failure of one or more sucker rods and, more often, in inefficient operation of the pumping system. It has not been possible, in the past, to predict the effect of changes in various operating variables of the system with any degree of accuracy, except by purely empirical or cut and try methods, for the obvious reason that the movement of the sucker rod string several miles below the surface of the earth cannot be observed nor is it possible to observe the manner of operation of the downhole pump. V

lt has been proposed to construct a mechanical model of the pumping system so that adjustments of various operating variables of the scale model would produce effects thereon similar to the effects of changes in corresponding variables upon the actual pumping system. However, in order to obtain any useful results by such a system, it can be demonstrated mathematically that the scale model itself would be of such large size and prohibitive cost as to render construction thereof entirely impractical. It has also been proposed to provide a system of weights and springs to behave in the same manner as the pumping system but, here again, the difficulties of adjustment of various parts of the system and the difficulties in making the model itself can be demonstrated to be so great as to eliminate the practical possibility of constructing such apparatus.

VI have discovered that a mechanical analogue of a pumping system can be set up on a practical scale to el1m1- nate the difficulties of providing a scale model or a system of weights and springs. In this system, a series of ilywheels connected by shafts are provided to represent the oil column, the tubing, and the sucker rod string, the rotary inertia of the flywheels representing the inertia of corresponding elements of the oil column, sucker rod string or tubing. The torque impressed upon the iiywheels represents stress imposed upon corresponding elements of the sucker rod string, tubing and oil column, while the angular velocity and displacement ofthe ilywheels represents the velocity and displacement ofcorresponding parts of the 2,698,133 APatented Dec. 28, k1954 pumping unit. In corresponding fashion, other parameters of the analogue or mechanical system represent variables and constants of the actual pumping unit. It is a feature of the invention that a clutch arrangement is provided to prevent simulation of tension upon the oil column so that the physical fact that oil cannot be subjected to tension is taken into account in constructing the analogue system.

lt is an object of the invention to provide an improved mechanical system which is an analogue of an actual well pumping unit, which system may be utilized to predict the effect of changing various operating conditions of the pumping system.

It is a still further object to provide apparatus which is reliable in operation, which utilizes standard mechanical components, and is of the lowest cost commensurate with the complexity of the data obtained therefrom.

Various other objects, advantages and features of the invention will become apparent from the following detailed description taken in conjunction with the accompanying drawings, in which:

Figure 1 is a vertical sectional View, partially in elevation, of a typical deep well pumping system;

Figure 2 is a view of the analogue system simulating the operation of the deep well pumping unit;

Figure 3 is a view of the pump-simulating unit representing conditions during the upstroke;

Figure 4 is a view of the pump-simulating unit representing conditions during the downstroke; and

Figures 5, 6, and 7 are diagrammatic views illustrating features of the invention.

Referring now to Figure 1, I have illustrated, in a schematic manner, a typical deep well pumping system to which the simulator of this invention is applicable, this system including driving mechanism 10 located at the surface and a downhole pumping unit 11. The unit 11 i11- cludes the usual well casing 12 within which is mounted tubing 13 having a sucker rod string 14 mounted therein. The string 14 includes a number of sucker rods 15 connected together by joints 16, the upper end of the string extending through a stuffing box 17 in a casing head 18 and the lower end being attached to a plunger 19 forming a part of a pump 20 which includes an upper or traveling valve 21 and a bottom or stationary valve 22. The uppermost sucker rod or polished rod 23 is driven by the mechanism 10 which includes a walking beam 24 having one end secured to the polished rod and its other end secured to a pitman 25 which is driven by a crank or counterbalance 26. The crank is driven by a flywheel 27, as through a chain drive 28, and the flywheel, in turn, is driven by a prime-mover 29, as by a second chain drive 30. The prime mover 29 may be an electric motor, an internal combustion engine, or any other suitable type of engine.

In accordance with the invention, the operation of the downhole unit 11 is mechanically simulated by the apparatus of my invention. The apparatus of Figure 2 simulates the behavior of the Vdownhole unit 11. The torque upon various parts of the analogue system represents stress at corresponding parts of the pumping unit, the angular displacement at various points of the analogue system represents displacement of corresponding parts of the actual pumping system, and angular velocity in the analogue system represents the velocity of corresponding parts of the mechanical system. In order for the two systems to be analogues of each other, it is necessary that the differential equations describing the behavior of the'analogue system be of similar form to the differential equations describing thebehavior of the actual pumping unit. Thereupon, by suitably adjusting the coefficients of various terms of the analogue system, as by changing the paramt eters of various parts of the analogue system, a condition O represents the oil column and a line R represents the sucker rod string 15'. The subscripts of the symbols repre sent the line to which they are applicable such as the tubing, oil or rod line. The following notation is used in the differential equations just referred to, each equation being given both in its dimensionless and regular form, the dimensionless variables being indicated by placing a bar over the symbol for the variable. It will be understood that each variable is thus represented by a pure number, indi cated by the barred letter, multiplied by an appropriate combination of fundamental units which, for the pumping unit, are time, length and force, and which, for the ana" logue system, are time, angle or angular displacement, and torque, convenient units for these variables being selected as set forth in the, rst part of the following table, the remainder of the table indicating the notation to be used hereinafter.

v-prime mover frequency or revolutions per second.

l-lielngth of segment of rod, tube or Unit of force-force required to stretch a sucker rod segment by an amount l.

h-positional constantfor engine.

wf, w. wirweightper unit length of sucker rod, oil and tube, respectivoly,

1441 w.i*weight o( plunger and valve, respectively.

k, k1, Liz-viscous, coecient in expressionsMu-l-v), kiwi-w), low giving viscous drag per unit length between sucker rod and oil,

betweenoil column and tube, and,

between tube and oil surrounding tube, respectively.

qn-npward pressure on bottom of tubedue to oil betwecn tube and t-time I, 11,., 2-ldisplaeement of' tube,

rod, and o il,`respectively.

um im, uhr-velocity of rod, oil, and

tube, respectively- (u,=vlii; 117=vlv w-V-vm).

a,.b, c=1fb, lj, 1in., res ctively, time derivatives ot ve oeity, or acceleration.

rterision of tubing line (r.=es)

un .IJ-amplitude of angular oscillation imposed by driving unit.

Unit of torque-torque required to twist segment of rod line through an angle ol oncradian.

f 1rG7'4 (E 2L Ef, En, Ei-torque to twist segment of rod, oil or tubing line through an angle of one radian.

M-equivalent moment oi inertia of driving unit.

F-equivalent drag coelhcient for simulator driving unit.

H- equivalent positional coeif`1- cient for driving unit.

Jr, J f., J i-constant torque imposed on lywheels in rod, oil and tubingv lines, respectively, simulating the weight of corresponding segments of these lines.

L, In, Ii-moment o inertia of f lywheels in rod, oil and tubing lines, respectively, simulating inertia of corresponding segments of rod, oil and tube.

J p, J i-constant torque imposed on flywheels simulating weight of plunger and valve, respectively.

K, K1, .K2-viscous coefficients of expressions simulating viscous drag, as corresponding to the expressions at the left.

Q-constant torque imposed on flywheel simulating upward pres` sure o oil on bottom of tube Q=EQ 1p1-moment of inertia of flywheel simulating inertia of plunger.

1v1-moment of inertia of flywheel simulating inertia of valve end andtubing` X, YM Z-angular displacement of iywheels simulating displace. mentofoil, rod, and tube, respectively.

Um Vn, Tlfn-angular velocity of fiywheels in rod, oil and tubing lines simulating velocity oi rod, oil epd tube..

AQ, Bn, C I= U, V, Wm respectively, time derivatives of veloclty, or acceleration;

.Pf-torque in shaft segment oi r od line simulating tension (P=EP).

(xn-.torque in shaft segment simulatmg gompression in oil line (Q=EQ).

R-t0rque in shaft segment simulatine tension of tubing line (R=ER).

G. r, L-shear modulus radius and length, respectively, of shaft inr rod, oil or tubing line.

In the mathematical analysis of the pumping unit and analogue system which follows, the respective differential equations are first derived by the method of finite differences and, from each such equation, the corresponding dimensionless equation is derived and follows the basic equation. in accordance with the method of finite differences, the casing, tubing and sucker rod string are divided into a large number N of segments. During operation of the pumping unit, each segment of the sucker rod moves upward or downward in accordance with the mechanical laws governing the system.

In particular, considering the nth segmenty of the sucker rod, as defined by the dotted lines 32, 33, Figure 5, in an interval of time At, a volume of rod sunAt passes out at the bottom of the segment, and a volume of rod sun-int passes into the top of the segment with the result that a volume Mun-Mimi is removed from the segment. Accordingly, in this interval of time At, there is an increment Ap in tensile stress in accordance with the following equation:

Eq. (I)

This equationhas its counterpart in the analogue system of Figure 2 wherein shaft` unit 34 represents the nth segment of the pumping unit, unit 35 represents the top of the pumping unit, and unit 36 represents the pump at the bottom of the well. The segment 34 of the rod line or shaft 37 includes a flywheel 38 having a moment of inertia lr simulating the inertia of the corresponding segment of the sucker rod, r1`he flywheel 3.8 is attached to one input of a differential 39, the other input of which is attached to a correspondingywheel of the O line or shaft to simulate viscous drag between the oil column and rod, the driven member of the differential being attached to a damper 40 which produces a resistive` torque K(Un-|-Vn). A constant torque Jr representing the weight of the rod segment is produced by attaching a string 41 to the iiy` wheel, the string in turn,4 being attached to a long spring 42 whose tension is not appreciably changed by movement of the flywheel, Alternatively, the liywheel may be the armature of a small electricmotor, which is subjected to a constant drager torque` by current and field adjustments to the motor. The inevitable rotational inertia of the differential may beapportioned between the two adjacent ywheels. In a time interval AT the nth ywheel 38 turns through an. angle UnAT, and the (n-l )th flywheel 38 turns through an. angle Un-inT. Accordingly, there is an increment of torque upon shaft 37 as set forth by the following equation:

It will be noted that Equations l and 2 are of identical;

form and, upon comparing the dimensionless forms of these equations, the following relationships are obtained betweenthe variables pn, un and tof thepumping unit and the. variablesv Pn, Un and T of the analogue system, the symbol fr denoting`v the timeratio and the symbol p denoting the stiffness ratio In the pumping unit of Figure l, the downward velocity of the rod at the nth station is un and the upward velocity of the oil is vn with the result that there is an upward viscous drag upon the rod element of magnitude Muri-vn); From thesedata, the equationiof motion of a rod segment 46 which straddles the segment between lines 32, 33 may be deduced, this segment extending distance above and below the nth station point. This segment has acting upon it a total tension p1t+1 on its bottom end, a total tension pn on its top end, the aforementioned upward viscous drag kl( un-l-vn) and a downward Weight wn. The resulting force produces a downward acceleration an at the mid-point of this segment in accordance with the equation Eq (6) In Figurev 2, a torque -Pn acts on one end of the nth segment and a torque Pn+1 acts on the other end of the segment. Spring 42 produces a constant torque Jr upon ilywheel 38 and differential 39 together with damper 40 produce a torque K(Un{ Vn). These torques accelerate the segment 34 in accordance with the following equation:

tem and the equations representing the pumping unit of Figure 1.

Similar equations may be derived with respect to the compressive strain of the segments of the oil column and the motion of a general segment of the oil column. Thus, in a time interval At, a volume sovnAt, Figure 6, of oil ows upwardly into the bottom of the nth segment, a volume sevaint flows out at the top of the nth segment and an increment s(vn-vn1)nt is accumulated within the seg- Vcrease Aqn in the total oil pressure of an amount determined by the following equation:

produces a viscous drag K1(Vn{-Wn) upon the oil line. yA` line 53 and a spring 55 produce a constant resistive torque J0 upon the ywheel 50. The following equation is derived similarly to Equation 2 and expresses the torque in shaft segment 51.

It will be noted that Equations 13 and 14 are of similar form and, upon comparing the coefficients of the dimensionless forms of the equations, the following relationships are obtained between the coecients andV variables `of the equations representing the analogue system and the equations representing the pumping unit.

qn=Qpn Eq. (15) E.a,ep Eq. (16) ment. This increment is compressed and produces an inv50 f 6 The equation of motion of the oil segment which eittends a distance above and below the lower end of the nth oil segment, as is shown by Figure 6, is governed by the viscous drag on `the oil due to the sucker rod movement which, as previously indicated, is k(u|v) and thev viscous drag on the oil due to the friction thereof against the tube, which is k1(vn+wn) both expressions representing drag per unit length. The bottom of this segment is subjected to the compressive force qn+1 and the top of this segment is subjected to the compressive force -qn. Finally, the weight of oil per unit length wo multiplied by the length l must be considered in the equation of motion which, therefore, becomes By analogy to the derivation of Equation 7, the equation of motion of segment 34 of the oil line is as follows:

It will be noted that Equations 17 and 18 are of similar form and, equating the coecients of the dimensionless forms of these equations, the following relationships are obtained between the coefficients and variables.

Jo=wolp (19) Io=wal3a2p Eq. (20) K1=kiz2p Eq. (21) Wn=UlWn Eq. (22) It is known that the oil column cannot support a tensile stress. That is, although the oil can be and is compressed, it cannot be subjected to tension. Therefore, Equation 17 is subject to the restriction that q, the stress on the oil column, cannot become negative which would represent a tensile stress. This important limitation has not been taken into account prior to my invention, insofar as I am aware, and this limitation destroys the continuity of differential Equation 17 which practically makes impossible a mathematical solution of the equation. How-r ever, this limitation may be readily imposed upon the apparatus of Figure 2 and, to this end, I have provided a clutch 56 at each segment where a shaft 51 joins a flywheel 50. Accordingly, the connection between the flywheel and shaft is released whenever the torque in these parts is in a sense corresponding to tension upon the oil column. This limitation is expressed mathematically by the following restrictions upon Equations 17 and 18.

The movement of the tubing is governed generally by the same equations as the movement of the sucker rod. Referring to Figure 7, and considering the nth segment of the tubing, as defined by dotted lines 60 and 6l, a volume of tubing stwnAt passes out of the bottom of the segment during an interval At, a volume of tubing stwn-iAt passes into the top of the segment and, accordingly, in this interval, there is an increment in tensile stress in accordance with the following equation.

This equation has its counterpart in the analogue system of Figure 2 in the segment 34 of the tubing line which includes a shaft 56a carrying a iiywheel 57 connected as previously stated to one side of differential 52. A constant torque It is impressed upon flywheel 57 by a line 58 connected to .a spring 59.. A damper 62 produces an addition variable torque upon the flywheel yrepresenting viscous drag KzWn upon those segments having oil between the tubing and casing. The torque upon shaft 56a is as follows:

snag-Wfmmr This equation `being of similar form to Equation 24, .the following relationships hold between lthe cocicients land variables of the dimensionless forms of the equations, .as previously set forth:

The `equation of motion Aof the nth tube segment which extends a 'distance In the analogue system `of Figure 2, the equation of motion is as follows, as will be evident upon comparison with 'Equation 7.

,Iton

' T he torque unit 62 is included only for those segments of the tubing wherein a stationary body of fluid is `positioned 'between the casing and tubing. The following relationships between the pumping unit and analogue system ,follow from comparison of the dimensionless forms of .Equations 28 and 29.

AvFrom the foregoing description, it will be evident that the simulator unit 34 is an analogue of any typical segment of the tubing, sticker rod string, and oil column. The actual pumping unit may be divided into as many segments as desired, each such segment being represented by .a corresponding unit v34 in the system of Figure 2, the accuracy becoming greater vwithan increasing number of segments. `In `order to compensate for the fact that a finite difference analysis is accurate only up to an interval of one-half the element of extension, half-flywheels I I o not shown, may be inserted at the ends of the respective rod, 4oil and tubing lines.

AS stated, the simulator as thus far `described simulates the sucker `rod string together with `its associated oil .column and tubing. The simulator also includes the unit 35 representing conditions at the top of the well and and Cit

the 1.unit 36 representing conditions at the bottom of the wel The top of the tubing string is anchored to the vcasing and to the `ground so that it must remain stationary, that is, its rdisplacement must be zero `at .all times. condition is represented in Figure 2 :by anchoring ythe tubing line, as indicated at 65, at its top. This condition is represented by the `following equations:

With respect to the oil line, lthe oil is discharged at the top of the tubing into la storage tank against -a substantially constant head. That is, the pressure at the top of the oil line is constant. This condition is realized in the system of Figure 2 by inserting a half flywheel 70 together with string and spring as shown by the parts 54 and 55 at the top of the oil line to simulate this end condition. This condition is represented by the following equations.

The sucker rod string is driven at its top end by the drive mechanism which includes the prime mover, flywheel, crank `and walking beam of Figure 1. Although I have disclosed `a simulator unit 4in .my copending application, Serial No. 204,926, filed January 8, 1951, now U. S. Patent No. 2,661,898, entitled Simulaton which takes into account the reaction of the sucker rod string upon the drive mechanism, which `unit will be connected through an electro-mechanical transducer to the top .of the rod line, for present purposes, the driving system may Ibe assumed to be a positively acting mechanical system vof one ldegree of freedom. The laction .of vthe drive mechanism upon the top of the sucker rod may1 therefore, be represented by the following function, the terms of which represent, respectively, the resultant force applied to the top of the sucker rod, the lumped kinetic energy of the drive mechanism, the lumped potential energy of the drive mechanism, the dissipative function of the engine and the force or reaction which the sucker rod exerts upon the drive mechanism.

For present purposes, it is assumed that the reaction of the sucker rod system upon the drive mechanism is negligible, which is true in certain practical applications and where results are desired only to a predetermined degree of accuracy, so that the motion at the top -end of the sucker rod string may be expressed as a sinusoidal function. As indicated, the reaction of the sucker rod system upon the drive mechanism is taken into account when the simulator for the drive mechanism herein before referred to is ,connected'rat the top end of the sucker rod line.

ln Figure 2, the driving unit is represented by mechanism 73, the .behavior of which is represented by the vfollowing equation:

zinf- LF- tI.- f Q- E A0+EU0+EY0+P0 Eq' Comparison between the Equations 36 and 37 yields the following relationships between the parameters of the analogue system and pumping unit:

M=mla2p Eq. (38) F=flap Eq (39) Y :mp Eq. (4o) The end conditions for the bottom end of the pumping unit are determined by the nature of the pump used,

its dimensions, and the material from which it is made.

ertia Ip of the plunger, this ywheel being secured to the oil line by a clutch 81 and a gear 82 secured to a flywheel 83 at the end of the cil line. When flywheel 80 rotates in a clockwise direction, corresponding to the upstroke of the pump, clutch 81 is engaged with the result that the rod line is mechanically connected to the oil line. That is VN=UN Eq. (41) l Furthermore, during the upstroke, the oil flowing into the working barrel 19, Figure 1, does not follow the plunger fast enough to exert appreciable pressure on its underside. The forces acting upon the plunger are, therefore, the weight of the plunger wp, the upward tension pN in the rod and the downward pressure qN of the oil above the plunger. The following equation, accordingly describes the motion of the plunger.

In Figure 3, when clutch 81 is engaged, a torque Pn is exerted upon tlywheel 80 by the rod line and a torque QN is trasmitted to this ywheel by gear 82. In addition a line 84 and spring 85 produce a constant torque Jp upon flywheel 80, the equation of motion of which is, therefore:

The following relationships follow from comparisonof the dimensonless forms of Equations 42 and 43.

Eq. (44) Eq. (45) The forces acting upon valve 22, Figure l, are its weight wb, the upward tension m in the metal of the tube and total upward pressure qb on the bottom of the valve due to the head of oil between the head and casing. 'ilhs equation of motion of the valve is, therefore, as

o ows:

wb-qb-r1v=wbcN/g esvl- 1*(1b/wt-u-b1N--g-cy Eq. (46) A tlywheel 86 of inertia Ib is secured to the end of the tubing line, the llywheel being subjected to a torque RN by the line. A constant torque (Jb-Q11) is also applied to the ywheel by a string or line 87 and a spring 88. The equation of motion of tlywheel 86 is, therefore:

Jb-Qb- R1-r=gbgAr v Qomparison of the dimensonless forms of these equations ylelds the following relationships:

Thus, during the upstroke, the assembly of Figure 3 produces a condition simulating the motion of and forces acting upon the plunger, traveling valve 21, and fixed valve 22 of Figure l.

During the interval of time representing the downstroke, unit 36 is arranged as shown in Figure 4. The downward velocities of plunger 19 and the end of tube 13 being uN `and wN, respectively, a` volume of oil (sr-I-so) (uN-wN)At is displaced upwardly through valve 21 during an interval of time At. This upwardly displaced oil occupies an equal volume sa(vN+uN)At in the column above the plunger. Accordingly,

In Figure 4,ywheel 80 is connected through a clutch 88a and gears 89 to one sideof a differential 90, the

WNtsJfs.)

This speed is added to the speed VN at clutch 93 to produce a rotation of gear unit 89. However, the gear ratio at 89 is such that the adjacent input to the diierential 1s s UN?.

Accordingly, the following relationships exist at the unit of Figure 4 when the shafts rotate in a counterclockwise direction, corresponding to the downstroke of the pump, the clutches 88a, 92 and 93 all being .engaged during this period.

UN=VN+WN is Eq- 52) So 8o If the dimensonless forms of Equations 51 and 52 are derived and their coefficients compared, certain of the same relationships already derived will be obtained between the coecients and variables of the two sets of equations. It will be further noted that the two equations are of similar form so that the unit of Figure 4 accurately simulates the aforesaid end condition during the downstroke period.

Since oil passes freely through the valve 21 during thev downstroke, the pressure intensity in the oil below and above plunger 19 is nearly equal. The cross-section of oil in the barrel, however, exceeds that above the plunger in the ratio and, consequently, the upward force on the lower end of the plunger exceeds that on the upper end of the plunger Further, the plunger is subjected to the downward force of its weight wp and the tension pN in the sucker rod string. The equation of motion of the plunger during the downstroke is, therefore,

Similarly, in Figure 4, assuming that the inertia of the differential 90 is distributed rotatably among the ily-n wheels 80 and 86, the torque at the lower side of y wheel 80 is It will be noted that Equations 53 and 54 are of the same form so that the simulator portrays the movement of the plunger during the downstroke. If the dimen sionless forms of these equations are derived and their coefficients are compared, certain of the relationships already obtained between the coecients and variables of the two systems will be found to hold.

Further during the downstroke period, the forces acting upon stationary valve 22 are the weight of the valve wb, the downward pressure qN of the oil in the barrel of the plunger, an upward pressure qb 0n the bottom of the valve head, and the tension rN in the metal at the end of the tube. This valve, accordingly, has the following equation of motion.

Il In Figurel 4, the torque at the upperend of ywheel 86 is Rrnthe torque atthe lower end of flywheel 86- is Q1v(sr+so), and a constant torque (Jb-Q10 is produced by springV 88 and line 87; The equation of motion of flywheel 86` is, therefore:

It will be noted that Equations 55 and 56 are of the same form andi, hence, the unit accurately simulates the operation of valve 22 during the downs-troke period. If thev dimensionless? forms of these equations are derived and their coeflicients compared, certain of their rela tionships already derived will be found.

In the operation of the circuit, the simulator of Figure 2 is used to simulate conditions existing in an actual pumping installation.. In, mis actual installation, most of! the. constants arey known` and these` constants areA uti.- lized by substituting` them. in, the equationsv previously derived'. to. determine the corresponding, parameters of similar parts ofthe analogues of these constants. Before making these substitutions, the value of u, v, and E are assumed from which data a and p may be readily calculated. In this connection it will.V be noted and that the cross sectionalv areas of various partsl of' the# actual pumpingv system aswell as their elastic modu'l lusY are: known from which it follows that pA is readily calculated` once a value for E is assumed; Itis furthernecessary to assume a value for the segment length lf' whichy may conveniently be the length of one section of. the. sucker' rod string and, i-n the particular unit shownby Figure 2, they equivalent driving forceI Q, and the maw inmm amplitude D` of the driving forceI are assumed quantities. When the simulator of my aforementiizmed1 copending application is utilized to produce the driving force, the quantities t2, n, and D are obtained by measurements of the simulator output. As an alternative, an actual model of. the engine shown in Eigure l as driv ing the sucker' rod may be used. A rack and pinion is then used to convert the' linear motion of polished rod into the rotation necessary for the rest of the simulator. Once these constants have been assumed and calculated, as stated, the known constants of the actual pumping system are substituted in the equations to provide corresponding. parameters for the constants of the analogue system. As an example. of suchsubstitution, referring. to Equation 8?, the' weight wr of a unit length of sucker rod is known and the actual pumping' system. is constructed. The quantities l, and p, are assumed or calculatedE asl just indicated and, upon substitution of these values` in: Equation 8., the quantity 1r, that is, the constaatsv of springs. 4Z' in the. rod segments are: readily cal cul'atcd. Similarly', with respect to- Equation. 9, since the weight of a rod segment wr is known and the quan tities l, n, and .p are known, or calculated, the value Ir of the inertia of the ywheel's 3S in the rod line are readily calculated from Equation 9. In an entirely similar fashion, the parameters of the analogue system set forth in Equations 10,. 1'6, 19', 20, 2Ll, 26, 27,. 3G, 31,. 33, 34, 35,. 38,` 39, 40,. 44, 45,. 48,. 49, and 50' are readily calculated. since the. righthand. side off each of these equa tions contains only the known or assumed quantities: l, u, Q, D, ,u., fr, .p, and g which are assumed or calculated in. the manner previously set forth together with a constant which is readily measurable or known on the actual pumping system. This enables' the entire unit of Figures 2, 3` and 4 to be` constructed, each component havingl a value as determined by the respective foregoing equations.

When this is done, the relationships' set forth in Equations 3, 4, 5f, 1l, l2, 15, 22 and 32 hold between the variables of the analogue system and the variables of the actual. pumping system.. The values of the assumed quantities are then changed until the analogue system behaves in a manner analogous to the actual pumping system. Thereupon, the effect of any changes desired in the parameters of the mechanical system may be accurately predicted by noting' the effect of corresponding changes on the analogue system. This enables operating conditions to be determined so that the dimensionsl andphysical conditions of the actual pumping system are at an. optimum to provide most eiiicient opera.- tion with resultant elimination of difficultiesl dueto sucker rod breakage and malfunctioning of other partsV of the apparatus;

Where all the necessary constants are not known in the actual system so that the correspondii'xg` parameters of the analogue system are correspondingly unknown, the. known values are used, in themanner previously set forth to determine the constants of the corresponding parts of the analogue circuit. Reasonable, values are assumed for the other known. constants: and corresponding values are setv in the simulator' byv use of the aforementioned.' equation..

The simulator and its operation are then studied and A conclusions are drawn@ as to theextenttowhich. the performanceV depends on: the. unknownj constants: for` whichV values had been assumed.. Usually it will? be found. that some of these are critical. Others may lie anywhere within a fairly wide range without appreciable effect on the performance of' the simulator; Also, operation of thev simulator suggests measurements to be taken on the prototype whicheither permit computation of thoseV critical` constants or serve as indices to the. performance of the prototype.

Finally, l` wish to point out that the driving unit 7?/ of Figure 2 can be theelectricaltA drivingmechanism sirnulator of my copending application, Serial No. 204,926, now Patent No. 2,661,898, a suitable mechano-electrical transducer being provided to convert the output of the drive mechanism simulator to. a mechanical impulse for driving the end of the rod line.

While` the invention. has been described. in connection with a present, preferred* embodiment thereof, it is to be understood that this description is illustrative only and. is not` intended to the.Y invention, theV scope of which isv defined" by ther appended claims;

I claim:

1'. In apparatus for mechanically simulating a downholle pumping System includingV a sucker rod string, tubing surrounding said sucker rodstring and` an oil. column between said rod and sai'd tube, in. combination;` a plurality of units, one for. eachV segment ot'. the. pumps ing system, each unit including a shaft representing each1 of a tubing segment, a rod segment, and an oil column: segment, each shaft having a flywheel coupled thereto' whose moment of inertia represents the inertia of the segment and a drag device coupled thereto representing the weight of the segment, means coupling the oil shaft and the. tubing shaft to simulate viscous drag between l the tubing segment and the oil column segment, means coupling the oil shaft and therod shaft to simulate viscous drag between the rod segment and. the oil cohmn. segment, and a clutch attached to each segment.: of the oil shaft to prevent the oil shaft torque from becoming negative;l means coupled to each of said. shafts for sirn-4 ulating the driving mechanism and end conditions at the top of the pumpingl system; and means coupled to each of said shafts simulating the operation of the pump at the bottom of said system.

2. In apparatus for mechanically simulating a downhole pumping systemA including al sucker rod string, tubing surrounding said sucker rod string and an oil column between said rod; and said tube, in combination? a plurality of units, one for each. segment of the pumping system, each unit including a shaft representing each of a tubing segment, a rod segment, and an oil column segment, each shaft having a flywheel coupled thereto whose moment of inertia represents the inertia of the segment and af drag device coupled thereto representing the weight of the segment, a. differential coupling, the oil, shaft and the tubing shaft tosimulatey viscous drag between they tubing segment and the oil column segment, and a dif ferential coupling the oil shaft and the rod shaft to simulate viscous drag between the rod segment and the oil column segment; means for simulating the driving. mechanism at the top of the pumping system including a drive device coupled to said rod shaft for producing a driving force representative of the force of the driving mechanism at the top of the pumping system, a constant torque device connected to the oil shaft to simulate. the pressure at which oil is discharged from the` tubing, and means secured to the top of the tubing shaft to prevent rotation thereof; and means coupled to each of said shafts for simulating the operation of the pump at the bottom of said system..

3. In apparatus for mechanically simulating a downhole pumping system including a sucker rod string, tubing surrounding said sucker rod string and an oil column between said rod and said tube, in combination; a plurality of units, one for each segment of the pumping system, each unit including a shaft representing each of a tubing segment, a rod segment, and an oil column segment, each shaft having a flywheel coupled thereto whose moment of inertia represents the inertia of the segment and a drag device coupled thereto representing the weight of the segment, a differential coupling the oil shaft and the tubing shaft to simulate viscous drag between the tubing segment and the oil column segment, and a differential coupling the oil shaft and the rod shaft to simulate viscous drag between the rod segment and the oil column segment; a unit for simulating the driving mechanism at the top of the pumping system including a drive device coupled to said rod shaft for producing a driving force representative of the force of the driving mechanism at the top of the pumping system, a constant torque device connected to the oil shaft to simulate the pressure at which oil is discharged from the tubing, and means secured to the top of the tubing shaft to prevent rotation thereof; and a unit for simulating the operation of the pump at the bottom of said system, said last-mentioned unit including a plurality of clutches, a ywheel at the end of each of the rod, oil and tubing shafts, a differential, a constant drag device attached to each of the last-mentioned ywheels in the rod and tubing shafts, said clutches coupling the rod shaft directly to the oil shaft during the period representing the upstroke of the pump, and said clutches coupling one side of the differential to the rod shaft, the other side of the differential to the tubing shaft, and the driven differential element to the oil shaft during the period representing the upstroke of the pump.

4. In apparatus for simulating the downstroke of a downhole pump including a plunger, a traveling valve, and a stationary valve, in combination, a shaft mechanically simulating each of the oil column, tubing and sticker' rod string; a unit for simulating the operation of the pump at the bottom of said system, said last-mentioned unit including a plurality of clutches, a flywheel attached to the end of each of the rod, oil and tubing shafts, a differential, a constant drag device connected to each of the last-mentioned fiywheels in the rod and tubing shafts. said clutches coupling the rod shaft directly to the oil shaft during the period representing the upstroke of the pump, and said clutches coupling one side of the differential to the rod shaft, the other side of the differential to the tubing shaft, and the driven differential element to the oil shaft during the period representing the upstroke of the pump.

5. Apparatus for simulating a downhole pumping unit comprising a shaft simulating a column of oil pumped by the unit; means for applying a plurality of torques to said shaft at spaced intervals thereon, said intervals representing individual segments of the oil column, said torques simulating stress on the oil column at corresponding segments thereof whereby said torques tend to cause rotation of said shaft simulating displacement of corresponding segments of the oil column; and means to disengage any one of the torques applied to said shaft whenever the value of said torque is such as to tend to cause rotation of said shaft in a direction which represents tension in the oil column.

6. The combination in accordance with claim wherein said means to disengage the torques from said shaft comprises a plurality of clutches coupling respective said means for applying torques to said shaft, said individual torques being connected to said shaft whenever the values thereof are such as to cause rotation of said shaft in a first direction and being disconnected from said shaft whenever the values thereof are such as to tend to cause rotation of said shaft in a second direction.

7. Apparatus for simulating a segment of a downhole pumping unit including a tubing segment, a rod segment, and an oil column segment comprising, in combination; first, second, and third shafts simulating, respectively, said tubing, rod, and oil column segments; first, second, and third flywheels attached, respectively, to said first, second, and third shafts, the inertia of said fiywheels simulating the inertia of corresponding segments; rst, second, and third constant drag means attached, respectively, to said first, second, and third shafts, said drag means simulating the weights of corresponding segments; means coupling said rod shaft to said oil column shaft whereby rotation of said rod shaft imparts rotation to said oil column shaft simulating viscous drag between the rod sebment and the oil column segment; and means coupling said oil column shaft to said tubing shaft whereby rotation of said oil column shaft imparts rotation to said tubing shaft simulating viscous drag between the oil column segment and the tubing segment.

8. The combination in accordance with claim 7 wherein said means coupling said rod shaft to said oil column shaft and said means coupling said oil column shaft to said tubing shaft each comprises a differential, the two driving members of said differential being connected, respectively, to said two shafts being coupled, and means coupled to the driven member of said differential to impart a drag thereto to simulate the viscous drag between corresponding segments in the pumping unit.

9. The combination in accordance with claim 7 further comprising means coupled to said tubing shaft to impart a viscous drag thereto tending to prevent rotation of said tubing shaft, said viscous drag representing the friction exerted by a stationary body of oil outside the tubing segment.

10. Apparatus for simulating a downhole pumping system including a sucker rod string, tubing surrounding said sucker rod string, an oil column between said rod and said tubing, drive mechanism positioned at the surface to impart reciprocating motion to the upper end of the sucker rod string, and a pump attached to the lower end of the sucker rod string to force said column of oil upward inside said tubing comprising, in combination; first, second, and third shafts simulating, respectively, said rod, tubing, and oil column; a plurality of first, second, and third flywheels attached, respectively, to each of said first, second, and third shafts, the inertia of said fiy wheels simulating the inertia of corresponding segments of said rod, tubing, and oil column; a plurality of first, second and third constant drag devices attached, respectively, to said first, second, and third shafts, said drag devices simulating the weight of corresponding segments of said rod, tubing, and oil column; means coupled to corresponding first ends of said rod, tubing, and oil shafts simulating the driving means and surface conditions of the pumping system, said last mentioned means imparting reciprocating rotary motion to said first end of said rod shaft; a plurality of means coupling said rod shaft to said oil column shaft whereby rotation of said rod shaft imparts rotation to said oil column shaft simulating viscous drag between respective rod segments and oil column segments; a plurality of means coupling said oil column shaft and said tubing shaft whereby rotation of said oil column shaft imparts rotation to said tubing shaft simu- 'lating viscous drag between respective oil column segments and tubing segments; and means coupled to corresponding second ends of said rod, tubing, and oil column shafts simulating the operation of the pump at the bottom of the system.

1l. The combination in accordance with claim l0 further comprising means coupled to said tubing shaft to impart a viscous drag thereto tending to prevent rotation of said tubing shaft, said viscous drag representing the friction exerted by a stationary body of oil outside the tubing segment.

l2. The combination in accordance with claim l0 wherein said means coupling the first ends of said shafts comprises a driving unit for imparting a reciprocating rotary motion to said first end of the rod shaft simulating the action of the driving unit attached to the uppermost end of the sucker rod string, means coupled to said first end of the oil column shaft to impart a constant torque thereto simulating the constant pressure at which the oil is discharged from the tubing, and means attached to said first end of the tubing shaft to prevent rotation thereof simulating rigid attachment of the tubing at the surface.

13. The combination in accordance with claim l0 wherein said means simulating operation of the pump positioned at the bottom of the system comprises a iiywheel connected to said second ends of each of the rod, tubing, and oil column shafts; means coupled to each of said last mentioned flywheels attached to the rod and tubing shafts toimpart a constant torque thereto; a differential; and a series of clutches, said clutches coupling 15 1,6 si wely shgifttq safidloilacglumn shaftlwhent/r the thrlfe References, Cited in the le of tlm',y patent s al.; rotate in. a rst` ,inaction simu ating ye upstrore of-j the pump,y and s aid clutches coupling one side of said UNITED STATES PATENTS A dijerential to said rod shaft, the o ther side of said dif- Number Name Date, fmentialj to said,v ltubing shaft, andA the movable differam 5v 1,228,392: Beighlee 1..- )une 5, 19,17 tialelgment tvo, sagid oil cgolumn` shaftgwhefnever said three 1,662,272 Klopstig Mar. 13 1928, shafts, rotate in asecond direction simukating the` down 1,799,1344 Hardy Mar, 31',V 193'1 sngoke. of the pump, 2,551,502 Montrose-Oster Y May 1v,- 1,951 

